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Home Vector Calculus Surface Area and Surface Integrals Examples Example 2
 
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Example 2

EXAMPLE 2

Find the area of the portion of the hyperbolic paraboloid z = x2 - y2 which is inside the cylinder x2 + y2 = 1.

The line integral has an analogue for surfaces called the surface integral. The form of the line integral which is most easily generalized to surfaces is the vector form

13_vector_calculus-273.gif

where N is the unit normal vector of C. This is convenient because surfaces also have unit normal vectors.

Step 1

Sketch the region (Figure 13.5.4).

13_vector_calculus-274.gif

Figure 13.5.4

Step 2

D is the region

13_vector_calculus-275.gif

or in polar coordinates,

0 ≤ θ ≤ 2π, 0 ≤ r ≤ 1.

Step 3

13_vector_calculus-276.gif

Then

13_vector_calculus-277.gif

It is easier to use polar coordinates, where

13_vector_calculus-278.gif

Put u = 4r2 + 1, du = 8r dr,

13_vector_calculus-279.gif

Home Vector Calculus Surface Area and Surface Integrals Examples Example 2

Last Update: 2006-11-22